Group decision making in the analytic hierarchy process by hesitant fuzzy numbers

Due to the increasing complexity of decision problems, many managers employ multiple experts to reach a good decision in a group decision making. Now, if there is ambiguity in the evaluation of experts, the use of fuzzy numbers can be useful for each expert. In these situations, the use of hesitant fuzzy numbers (HFNs) which consists of several fuzzy numbers with special conditions can be suggested. HFNs are as an extension of the fuzzy numbers to take a better determining the membership functions of the parameters by several experts. Because of simple and fast calculations, in this paper, we use triangular HFNs in the pairwise comparison matrix of analytic hierarchy process by opinions of a group of decision makers in a hesitant fuzzy environment. We define consistency of the hesitant fuzzy pairwise comparison matrix and use the arithmetic operations on the HFNs and a new method of comparing HFNs to get the hesitant fuzzy performance score. By using score function to hesitant fuzzy score we can get a final score for alternatives. Finally, a practical example is provided to show the the effectiveness of this study. The obtained results from this paper show that new method can get a better answer by keeping the experts’ opinions in the process of solving the problem.


Group decision making in the analytic hierarchy process by hesitant fuzzy numbers
Mahdi Ranjbar 1* & Sohrab Effati 1,2 Due to the increasing complexity of decision problems, many managers employ multiple experts to reach a good decision in a group decision making.Now, if there is ambiguity in the evaluation of experts, the use of fuzzy numbers can be useful for each expert.In these situations, the use of hesitant fuzzy numbers (HFNs) which consists of several fuzzy numbers with special conditions can be suggested.HFNs are as an extension of the fuzzy numbers to take a better determining the membership functions of the parameters by several experts.Because of simple and fast calculations, in this paper, we use triangular HFNs in the pairwise comparison matrix of analytic hierarchy process by opinions of a group of decision makers in a hesitant fuzzy environment.We define consistency of the hesitant fuzzy pairwise comparison matrix and use the arithmetic operations on the HFNs and a new method of comparing HFNs to get the hesitant fuzzy performance score.By using score function to hesitant fuzzy score we can get a final score for alternatives.Finally, a practical example is provided to show the the effectiveness of this study.The obtained results from this paper show that new method can get a better answer by keeping the experts' opinions in the process of solving the problem.
It is difficult for an expert to be able to consider all aspects of a decision-making problem.Therefore, group decision-making would often be preferred and would generate more benefits than individual decision-making.The relationships among the decision makers are important factors that affect on group decision-making process 1 .Also, if they are like-minded, they are aligned in choosing their opinions, but they may have hesitance in choosing the membership function as a fuzzy number in different forms.In most research articles on group decisionmaking, the opinions of different decision makers are aggregated, which causes the loss of some information.In such a situation, using a new approach can be useful.In this article, we try to solve this problem by considering the extension of fuzzy numbers and using the existing arithmetic operations on them.
In the theory of decision making, the analytic hierarchy process (AHP) is a structured technique for organizing and analyzing complex decisions.It was developed by Saaty 2 , which the experts usually provide crisp values for decisions over paired comparisons of alternatives with respect to a criterion.If the experts are uncertain on the decisions, this uncertainty can be measured by intervals 3 .In uncertain situations, the decisions can also be represented by fuzzy values.As a popular methodology for confronting with uncertainty, the fuzzy logic combined with AHP, more commonly known as fuzzy AHP (FAHP), has found more applications in recent years 4 .Laarhoven and Pedrycz 5 presented a fuzzy version of AHP method.Buckley used fuzzy priorities of comparison ratios in place of exact ratios 6 .Chang introduced a new approach for FAHP with using triangular fuzzy numbers in pairwise comparison scale 7 .Cheng presented a new approach for evaluating naval tactical missile systems depending by the FAHP 8 .Chan and Kumar used fuzzy extended AHP-based approach to global supplier development considering risk factors.Huang et al. presented a FAHP method and utilize crisp judgment matrix to evaluate subjective expert judgments made by the technical committee of the Industrial Technology Development Program in Taiwan 9 .Tang provided an efficient budget allocation method using FAHP for businesses 10 .Das et al. focused on performance evaluation and ranking of seven Indian institute of technology in respect to stakeholders' preference using an integrated model consisting of FAHP and compressed proportional assessment methods 11 .Deng applied a FAHP approach for tackling qualitative multi criteria analysis problems 12 .Cheng et al. considered attack helicopters based on linguistic variables by a FAHP method 13 .Leung and Cao proposed a fuzzy consistency of a tolerance deviation in the FAHP method 14 .Karczmarek et al. developed FAHP in a graphical approach.
decision making system for computer numerical control router selection 69 .Tuysuz and Simsek have also benefited HFLTS based on HFAHP in order to evaluate performance of the logistics firm which has 1000 branches in Turkey 70 .Buyukozkan and Guler proposed an supply chain analytics tool evaluation model by using HFLTS and AHP method 71 .Samanlioglu et al. applied HFAHP to measurement of the COVID-19 pandemic intervention strategies 72 .Candan and Toklu solved the most appropriate location problem for the solar power plant by HFAHP method 73 .Batur Sir and Sir used an HFLTS in the AHP and VIKOR method to treat the pain symptoms observed in COVID-19 patients 74 .Candan and Cengiz determined solar power plant location using HFAHP method 73 .
In most of these studies for HFAHP method used HFSs or hesitant fuzzy linguistic term sets to select elements of hesitant fuzzy pairwise comparison matrix (HFPCM), which usually solve these problems by aggregation the opinions of decision makers.In this paper, we want to use these elements on a special type of the HFNs, which creates a new form for using AHP method in a hesitant fuzzy environment.Then, the required definitions and theorems have been prepared and a new algorithm introduced to rank of alternatives by AHP method in these conditions.One of the advantages of this approach is that the HFNs are effective on arithmetic operations, similar to the fuzzy numbers in fuzzy mathematics, that cause to reduce the volume of calculations and to apply the expertise of decision makers in all problem-solving processes.
The remainder of this paper has been formed as follows: In section "Preliminaries", we provide some needed definitions and notions.In section "Algorithm of new approach for HFAHP", we propose an algorithm to solve HFAHP method.In section "Illustration", one example to illustrate of the proposed algorithm is provided.The comparative analysis is done in section "Comparative analysis".A discussion is given in section "Discussion".Finally, some conclusions and recommendations for future research are discussed in section "Conclusion".

Preliminaries
Definition 1 75 If S is a collection of objects denoted by s, then a fuzzy set F in S is a set of ordered pairs which µ F (s) is entitled the membership function of x in F.
Fuzzy numbers are a type of fuzzy sets that on the set R under special conditions are defined.Triangular and trapezoidal fuzzy numbers are often used to sake of computational efficiency.A trapezoidal fuzzy number is a fuzzy number represented with quaternary notation as F = (f 1 , f 2 , f 3 , f 4 ) , this representation is interpreted as membership function as follows: Also, if in the quaternary notation (f 1 , f 2 , f 3 , f 4 ) we have f 2 = f 3 , then we can be represented it by the ternary notation (f 1 , f 2 , f 4 ) as a triangular fuzzy number (TFN).
In the next definition, we introduce the HFSs.
Definition 2 33 Let Y be a reference set which its objects defined by y; then the HFS F on Y is defined as a set of ordered pairs as follows: where h F (y) = {f 1 , . . .f l(y) } with l(y) = |h F (y)| is the possible membership degrees of the element y ∈ Y to the set F .For convenience, h F (y) is named a hesitant fuzzy element (HFE).
Definition 3 76 For an HFE h F (y) , S(h , where l(y) is the cardinality of h F (y).Some operations on two HFE h 1 and h 2 and ∈ R + are defined in 76 as follows: y , denote an HFS, which Y is infinite and h F (y) = {µ F1 (y), . . .µ Fl(y) (y)}.
Definition 4 77 A HFS Ũ on Y is defined uniformly HFS (UHFS) if there is a number p such that l(y) ≤ p for each y ∈ Y.

Characteristic of the each element of UHFS Ũ defined as
we express the UHFS Ũ with Ũ = { Ũj , } p j=1 , while l(y) = p for all y ∈ Y.In the following the definition of an HFN is introduced.Definition 5 78 Let Ẽ be a UHFS as follows: Then we named it an HFN, if where Ẽj 1 is an 1-cut for the jth ( j = 1, . . ., p ) element of the UHFS Ẽ and F is space of fuzzy numbers.The space of HFNs denoted with HF .One reason for using HFNs is that in some of decision-making prob- lems, all experts agree on a fuzzy number as linguistic value for a attribute of the alternative, but disagree on the choice of the hedge term for that item.For example, when experts evaluate the 'Design' of a car, linguistic labels like 'Good' , 'Fair' and 'Weak' are usually used.Let for label 'Fair' , all experts agree with the fuzzy number ' 5 ' , but there is a difference in determining its hedges.In such situations we propose to use of the HFNs. Figure 1 shows various hedges for the fuzzy number ' 5 ' by four decision makers.www.nature.com/scientificreports/for all z ∈ R , and let σ : (1, . . ., p) → (1, . . ., p) be a permutation, where Gσ(j) and Hσ(j) are the jth smallest membership function in HFNs G and H , respectively, that ordered by a ranking function as Yager index 79 .
It should be noted that, for more convenience in calculations in practice, the HFN T1 ⊗ ˜T 2 which its elements are not necessarily TFNs and can be approximated by the TFNs using the hypothesis of left and right divergence 80 and it call an approximation of the given HFN.
For definition of consistency in hesitant fuzzy pairwise comparison matrix (HFPCM), we need to introduce a method of comparing HFNs.Thus, by extension principle on HFSs we give the definition as follows.
Definition 9 Let G = { Gσ(l) } p l=1 and H = { Hσ(l) } p l=1 be two HFNs, We will say that G and H are approximately equal which is written as G ≅ H , if G is not greater than H and H is not greater than G.
The crisp pairwise comparisons matrix A = [a ij ] m×m consistent if only if a ij = a ik a kj for all i, j and k.Based on Definition 9 we define consistency for HFPCM Ã as follows.
Theorem 1 Let B = [ Tij ] m×m is a hesitant fuzzy positive symmetric matrix, where Proof According to Theorem 2.1 in 6 , for each l = 1, . . ., p we have and thus from ( 3) and ( 4 Corollary 1 For the crisp pairwise comparisons matrix B = [T ij ] , the Consistency Ratio (CR) of B is defined as where the consistency index of B is given by max − n n − 1 , which max is the largest eigenvalue of B and the random index refers to the average consistency of randomly generated matrices of certain order, whose elements are chosen on 9-point scale.The solution to the AHP is acceptable only when the CR is less than or equal to 0.10 for all pairwise comparison matrices 2 .According to Theorem 1, for an HFAHP with HFPCMs { Bl = [ Tl ij ]} t l=1 , the solution to the HFAHP is acceptable only when the CR is less than or equal to 0.10 for all crisp pairwise comparison matrices as

Algorithm of new approach for HFAHP
In this algorithm for handling HFAHP method, we use the THFNs in pairwise comparison scale as a extension of the extent analysis method on FAHP in 7 .Based on, the new HFAHP method can be described as shown below in four algorithmic steps. (3) Table 1.The HFPCM of performance alternatives with the THFNs.www.nature.com/scientificreports/ Step 1: The experts determine the relative importance of each pair in pairwise comparisons matrix with THFNs.Table 1 shows the HFPCM of performance of n alternatives.
Where, Tij = { Tσ(l) ij } p l=1 represents the evaluations of p experts on comparison of i-th element to j-th element in a hesitant environment as THFN.It should be noted, since for each i = 1, 2, . . ., m , importance of A i over A i is exactly equal, then Tii = {(1, 1, 1)} .Table 2 shows the linguistic terms that are transformed into THFNs.
For example, in Table 2 is representative worth of element i over element j under evaluation of p experts, which L k and U be greater, then it show more uncertainty in the opinion of the l-th expert.Note, if Tij is representative worth of element i over element j, then Tji = ( Tij ) −1 .
Step 2: After formation HFPCMs with HFNs in Step 1, in this step we examine the acceptability of consistency of them by using corollary of Theorem 1.
Step 3: In this step, the amount of hesitant fuzzy synthetic extent for the i-th object of the HFPCM [ Tij ] m×m , is obtained as are THFNs, which are obtained as follows: and Then, based on we assign weight of i-th agent in the HFPCM Ã as follows: for each j = 1, . . ., m and j = i .It should be noted that the following three conditions are considered to determine h( Si ≥ Sj ) : i) If S i ≥ S j then h σ (1) ( Si ≥ Sj ) = 1 , for each l ∈ {1, . . ., p}.
ii   www.nature.com/scientificreports/ Step 4: In this step, we take the hesitant fuzzy performance score of each alternative.For this purpose we use operations which are introduced in 76 on HFEs.Based on, we can aggregate evaluations of all the experts by score functions in Definition 3 as a crisp decision-making.

Illustration
For the verification of the proposed HFAHP algorithm one example is selected.
Example 2 In a university assume that the post of a professor is vacant, and three volunteers V 1 , V 2 and V 3 remain.A committee has convened to choose the best possible volunteer for the vacant post.The committee has two members and they have identified the following attributes for this selection: .
Table 3. HFPCM for attributes.www.nature.com/scientificreports/this approach can be to get of hesitant fuzzy scors.Also, in cases where decision-making is more sensitive and the use of methods with aggeregation opinions have no results, this new approach can be beneficial.It should be noted that the use of other types of fuzzy numbers can be used in the opinion of experts, but since this causes the complexity and high volume of calculations, we use THFNs that are very common and understandable.Generally, using of the HFNs and their applications in optimization and decision-making problems can be beneficial in maintaining problem information.

Conclusion
This paper has shown how HFNs can be used to AHP method in the hesitant fuzzy environments.To solve this problem, at first we introduce a comparative method for two HFNs by extension principle on HFSs, then by it we investigated consistency in the HFPCMs.Finally we propose a new algorithm for HFAHP with THFNs that it gives a hesitant fuzzy performance score for ranking alternatives.It should be noted that due to the characteristics of HFNs and easier calculations on them, in future studies they can be used in other methods for AHP such as eigenvector, geometric mean and other decision-making methods in a hesitant fuzzy environments, in spite of some limitations they may have.

Limitations of the proposed work
One of the limitations that can be considered for this approach, it is that the form of the some experts' opinions may not be in the form of an HFN and provide as an HFS.Also, another limitation is to use the triangular HFNs form.Since it is common and effective to use triangular fuzzy numbers in fuzzy environments, we have considered this type of numbers for calculations.Although using other forms of HFNs can be used in this paper that it increases the volume and time of calculations and some definitions and formulas must be changed.However, the related challenges can be more, which we will address in future studies.

Future work
In the future, we will be focusing on developing new approach for other methods in decision making such as TOPSIS, VIKOR, Best-Worth, ... that some of them may provide new definitions such as distance in this space.Also, the relationship between HFSs especially HFNs and other extensions of fuzzy sets such as intuitive fuzzy sets, neurosophic sets, soft sets, and the use of their combination in solving decision problems can be studied as the next future works. Vol

Definition 6 5 Figure 1 .
Figure 1.Various hedges for the fuzzy number ' 5′ by four decision makers.

Table 2 .
Linguistic scale for the HFAHP with the THFNs.

Table 15 .
Fuzzy normalized weights of attribute.

Table 16 .
Fuzzy normalized weights of volunteers under each attribute.